Kissing numbers of regular graphs

TL;DR

This paper establishes a sharp upper bound on the number of shortest cycles in connected graphs, characterizes Moore graphs through equality conditions, and demonstrates super-linear kissing numbers in certain Ramanujan graphs.

Contribution

It provides a new characterization of Moore graphs via cycle bounds and improves existing inequalities for regular graphs, also analyzing kissing numbers in Ramanujan graphs.

Findings

Sharp upper bound on shortest cycles in graphs

Equality characterization of Moore graphs

Super-linear kissing numbers in Ramanujan graphs

Abstract

We prove a sharp upper bound on the number of shortest cycles contained inside any connected graph in terms of its number of vertices, girth, and maximal degree. Equality holds only for Moore graphs, which gives a new characterization of these graphs. In the case of regular graphs, our result improves an inequality of Teo and Koh. We also show that a subsequence of the Ramanujan graphs of Lubotzky-Phillips-Sarnak have super-linear kissing numbers.